plz help this question....?

Question

The side a of a triangle ABC is calculated from the following formula
a/(sin(A) = b/sin(B)
where lower-case letters refer to the sides and upper-case letters refer to the angles opposite those sides.
The side b may be in error by 1.1%.
A and B are given as 25 degrees and 58 degrees respectively, to the nearest degree.
Using partial differentiation find the greatest percentage error in the calculated value of a.

Answer 1

Differentiation isn't the best way to solve this, because it can just be solved directly.

a / sin(A) = b / sin(B) ==> a = b*sin(A) / sin(B)

So using the nominal values, a = b*sin(25) / sin(58) = 0.498b. Within the given error, a is maximized as a = 1.011b*sin(26) / sin(57) = 0.528b. a is minimized as a = 0.989b*sin(24) / sin(59) = 0.469b. The maximum value differs more from the nominal, so the percentage error is 100*(0.528 - 0.498) / 0.498 = 6.04%.

If you need to use differentiation, we take a = b*sin(A) / sin(B) and partially differentiate with respect to A and (separately) B. Wee see that a depends on cos(A) and -cos(B) / sin^2(B). If A and B each differ by 1 degree, the total variation is 1*cos(A) + 1*-cos(B) / sin^2(B) = cos(25) - cos(58) / sin^2(58) = 0.169. The variation with respect to b is linear, so we can just multiply this by 1.1% and divide by the coefficient of b in the nominal value to get 0.169*0.011 / 0.498 = 0.374%, which is not accurate. This is probably because a linear approximation is poor for this function.